More Time-Space Tradeoffs for Finding a Shortest Unique Substring

We extend recent results regarding finding shortest unique substrings (SUSs) to obtain new time-space tradeoffs for this problem and the generalization of finding k-mismatch SUSs. Our new results include the first algorithm for finding a k-mismatch SUS in sublinear space, which we obtain by extending an algorithm by Senanayaka (2019) and combining it with a result on sketching by Gawrychowski and Starikovskaya (2019). We first describe how, given a text T of length n and m words of workspace, with high probability we can find an SUS of length L in O(n(L/m)logL) time using random access to T, or in O(n(L/m)log2(L)loglogσ) time using O((L/m)log2L) sequential passes over T. We then describe how, for constant k, with high probability, we can find a k-mismatch SUS in O(n1+ϵL/m) time using O(nϵL/m) sequential passes over T, again using only m words of workspace. Finally, we also describe a deterministic algorithm that takes O(nτlogσlogn) time to find an SUS using O(n/τ) words of workspace, where τ is a parameter

A framework for space-efficient string kernels

String kernels are typically used to compare genome-scale sequences whose length makes alignment impractical, yet their computation is based on data structures that are either space-inefficient, or incur large slowdowns. We show that a number of exact string kernels, like the $k$-mer kernel, the substrings kernels, a number of length-weighted kernels, the minimal absent words kernel, and kernels with Markovian corrections, can all be computed in $O(nd)$ time and in $o(n)$ bits of space in addition to the input, using just a $\mathtt{rangeDistinct}$ data structure on the Burrows-Wheeler transform of the input strings, which takes $O(d)$ time per element in its output. The same bounds hold for a number of measures of compositional complexity based on multiple value of $k$, like the $k$-mer profile and the $k$-th order empirical entropy, and for calibrating the value of $k$ using the data

Linear-time String Indexing and Analysis in Small Space

The field of succinct data structures has flourished over the past 16 years. Starting from the compressed suffix array by Grossi and Vitter (STOC 2000) and the FM-index by Ferragina and Manzini (FOCS 2000), a number of generalizations and applications of string indexes based on the Burrows-Wheeler transform (BWT) have been developed, all taking an amount of space that is close to the input size in bits. In many large-scale applications, the construction of the index and its usage need to be considered as one unit of computation. For example, one can compare two genomes by building a common index for their concatenation and by detecting common substructures by querying the index. Efficient string indexing and analysis in small space lies also at the core of a number of primitives in the data-intensive field of high-throughput DNA sequencing. We report the following advances in string indexing and analysis: We show that the BWT of a string T is an element of {1, . . . , sigma}(n) can be built in deterministic O(n) time using just O(n log sigma) bits of space, where sigma We also show how to build many of the existing indexes based on the BWT, such as the compressed suffix array, the compressed suffix tree, and the bidirectional BWT index, in randomized O(n) time and in O(n log sigma) bits of space. The previously fastest construction algorithms for BWT, compressed suffix array and compressed suffix tree, which used O(n log sigma) bits of space, took O(n log log sigma) time for the first two structures and O(n log(epsilon) n) time for the third, where. is any positive constant smaller than one. Alternatively, the BWT could be previously built in linear time if one was willing to spend O(n log sigma log log(sigma) n) bits of space. Contrary to the state-of-the-art, our bidirectional BWT index supports every operation in constant time per element in its output.Peer reviewe

r-indexing the eBWT

The extended Burrows-Wheeler Transform (eBWT) [Mantaci et al. TCS 2007] is a variant of the BWT, introduced for collections of strings. In this paper, we present the extended r-index, an analogous data structure to the r-index [Gagie et al. JACM 2020]. It occupies O(r) words, with r the number of runs of the eBWT, and offers the same functionalities as the r-index. We also show how to efficiently support finding maximal exact matches (MEMs). We implemented the extended r-index and tested it on circular bacterial genomes and plasmids, comparing it to five state-of-the-art compressed text indexes. While our data structure maintains similar time and memory requirements for answering pattern matching queries as the original r-index, it is the only index in the literature that can naturally be used for both circular and linear input collections. This is an extended version of [Boucher et al., r-indexing the eBWT, SPIRE 2021]

Computing MEMs and Relatives on Repetitive Text Collections

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern $P[1 .. m]$ on a large repetitive text collection $T[1 .. n]$, which is represented as a (hopefully much smaller) run-length context-free grammar of size $g_{rl}$. We show that the problem can be solved in time $O(m^2 \log^\epsilon n)$, for any constant $\epsilon > 0$, on a data structure of size $O(g_{rl})$. Further, on a locally consistent grammar of size $O(\delta\log\frac{n}{\delta})$, the time decreases to $O(m\log m(\log m + \log^\epsilon n))$. The value $\delta$ is a function of the substring complexity of $T$ and $\Omega(\delta\log\frac{n}{\delta})$ is a tight lower bound on the compressibility of repetitive texts $T$, so our structure has optimal size in terms of $n$ and $\delta$. We extend our results to several related problems, such as finding $k$-MEMs, MUMs, rare MEMs, and applications

Motif Discovery with Compact Approaches - Design and Applications

In the post-genomic era, the ability to predict the behavior, the function, or the structure of biological entities, as well as interactions among them, plays a fundamental role in the discovery of information to help biologists to explain biological mechanisms. In this context, appropriate characterization of the structures under analysis, and the exploitation of combinatorial properties of sequences, are crucial steps towards the development of efficient algorithms and data structures to be able to perform the analysis of biological sequences. Similarity is a fundamental concept in Biology. Several functional and structural properties, and evolutionary mechanisms, can be predicted comparing new elements with already classified elements, or comparing elements with a similar structure of function to infer the common mechanism that is at the basis of the observed similar behavior. Such elements are commonly called motifs. Comparison-based methods for sequence analysis find their application in several biological contexts, such as identification of transcription factor binding sites, finding structural and functional similarities in proteins, and phylogeny. Therefore the development of adequate methodologies for motif discovery is of paramount interests for several fields in computational biology. In motif discovery in biosequences, it is common to assume that statistically significant candidates are those that are likely to hide some biologically significant property. For this purpose all the possible candidates are ranked according to some statistics on words (frequency, over/under representation, etc.). Then they are presented in output for further inspection by a biologist, who identifies the most promising subsequences, and tests them in laboratory to confirm their biological significance. Therefore, when designing algorithms for motif discovery, besides obviously aim at time and space efficiency, particular attention should be devoted to the output representation. In fact, even considering fixed length strings, the size of the candidate set become exponential if exhaustive enumeration is applied. This is already true when only exact matches are considered as candidate occurrences, and worsen if some kind of variability (for example a fixed number of mismatches is allowed). Alternatively, heuristics could be used, however without the warranty of finding the optimal solution. Computational power of nowadays computers can partially reduce these effects, in particular for short length candidates. However, if the size of the output is too big to be analyzed by human inspection the risk is to provide biologists with very fast, but useless tools. A possible solution relies on compact approaches. Compact approaches are based on the partition of the search space into classes. The classes must be designed in such a way that the score used to rank the candidates has a monotone behavior within each class. This allows the identification of a representative of each class, which is the element with the highest score. Consequently, it suffices to compute, and report in output, the score only for the representatives. In fact, we are guaranteed that for each element that has not been ranked there is another one (the representative of the class it belongs to) that is at least equally significant. The final user can then be presented with an output that has the size of the partition, rather than the size of the candidate space, with obvious advantages for the human-based analysis that follows the computer-based filtering of the pattern discovery algorithm. Compact approaches find applications both in searching and discovery frameworks. They can also be applied to several motif models: exact patterns, patterns with given mismatch distribution, patterns with unknown mismatch distribution, profiles (i.e. matrices), and under both i.i.d. and Markov distributions. The purpose of this chapter is to describe the basis of compact approaches, to provide the readers with the conceptual tools for applying compact approaches to the design of their algorithm for biosequence analysis. Moreover, examples of compact approaches that have been successfully developed for several motif models (e.g. exact words, co-occurrences, words with mismatches, etc) will be explained, and experimental results to discuss their power will be presented